Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\protect \mathbb{R}^N$

Christos Sourdis

doi:10.5802/crmath.150

Équations aux dérivées partielles

Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in

ℝ^{N}

Christos Sourdis ¹

¹ National and Kapodistrian University of Athens, Department of Mathematics, Athens, Greece.

Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 131-136.

Résumé

We show that the elliptic problem $Δ u + f (u) = 0$ in $ℝ^{N}$ , $N \geq 1$ , with $f \in C^{1} (ℝ)$ and $f (0) = 0$ does not have nontrivial stable solutions that decay to zero at infinity, provided that $f$ is nonincreasing near the origin. As a corollary, we can show that any two nontrivial solutions that decay to zero at infinity must intersect each other, provided that at least one of them is sign-changing. This property was previously known only in the case where both solutions are positive with a different approach. We also discuss implications of our main result on the existence of monotone heteroclinic solutions to the corresponding reaction-diffusion equation.

Reçu le : 2020-08-26
Révisé le : 2020-11-13
Accepté le : 2020-11-15
Publié le : 2021-03-17

DOI : 10.5802/crmath.150

Affiliations des auteurs :

Christos Sourdis ¹

¹ National and Kapodistrian University of Athens, Department of Mathematics, Athens, Greece.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Christos Sourdis},
     title = {Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\protect \mathbb{R}^N$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {131--136},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {2},
     year = {2021},
     doi = {10.5802/crmath.150},
     language = {en},
}

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Christos Sourdis. Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\protect \mathbb{R}^N$. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 131-136. doi : 10.5802/crmath.150. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.150/

Bibliographie
Cité par

[1] Giovanni Alberti; Luigi Ambrosio; Xavier Cabré On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., Volume 65 (2001) no. 1-3, pp. 9-33 | DOI | MR | Zbl

[2] Henri Berestycki; Luis Caffarelli; Louis Nirenberg Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 25 (1997) no. 1-2, pp. 69-94 | Numdam | MR | Zbl

[3] Henri Berestycki; Louis Nirenberg On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat., Nova Sér., Volume 22 (1991) no. 1, pp. 1-37 | DOI | MR | Zbl

[4] Henri Berestycki; Louis Nirenberg; S. R. Srinivasa Varadhan The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Commun. Pure Appl. Math., Volume 47 (1994) no. 1, pp. 47-92 | DOI | MR | Zbl

[5] Jérôme Busca; Mohamed Ali Jendoubi; Peter Poláčik Convergence to equilibrium for semilinear parabolic problems in $ℝ^{N}$ , Commun. Partial Differ. Equations, Volume 27 (2002) no. 9-10, pp. 1793-1814 | DOI | MR | Zbl

[6] Xavier Cabré Uniqueness and stability of saddle-shaped solutions to the Allen–Cahn equation, J. Math. Pures Appl., Volume 98 (2012) no. 3, pp. 239-256 | DOI | MR | Zbl

[7] Xavier Cabré; Antonio Capella On the stability of radial solutions of semilinear elliptic equations in all of $ℝ^{n}$ , C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 10, pp. 769-774 | DOI | MR | Zbl

[8] Xavier Cabré; Alessio Figalli; Xavier Ros-Oton; Joaquim Serra Stable solutions to semilinear elliptic equations are smooth up to dimension $9$ , Acta Math., Volume 224 (2020) no. 2, pp. 187-252 | DOI | MR

[9] Edward Norman Dancer Stable and finite Morse index solutions on $ℝ^{n}$ or on bounded domains with small diffusion, Trans. Am. Math. Soc., Volume 357 (2005) no. 3, pp. 1225-1243 | DOI | MR | Zbl

[10] G. H. Derrick Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys., Volume 5 (1964), pp. 1252-1254 | DOI | MR

[11] Louis Dupaigne Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 143, CRC Press, 2011 | MR | Zbl

[12] Louis Dupaigne; Alberto Farina Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains (2019) (https://arxiv.org/abs/1912.11639)

[13] Alberto Farina Liouville-Type Theorems for Elliptic Problems, Handbook of Differential Equations 4: Stationary Partial Differential Equations, Elsevier, 2007, pp. 61-116 | Zbl

[14] Changfeng Gui; Wei-Ming Ni; Xuefeng Wang On the stability and instability of positive steady states of a semilinear heat equation in $ℝ^{n}$ , Commun. Pure Appl. Math., Volume 45 (1992) no. 9, pp. 1153-1181 | MR | Zbl

[15] François Hamel; Hirokazu Ninomiya Localized and expanding entire solutions of reaction-diffusion equations (2020) (https://arxiv.org/abs/2005.07420)

[16] Yi Li; Wei-Ming Ni Radial symmetry of positive solutions of nonlinear elliptic equations in $ℝ^{n}$ , Commun. Partial Differ. Equations, Volume 18 (1993) no. 5-6, pp. 1043-1054 | DOI | MR | Zbl

[17] Chia-Ven Pao Nonlinear Parabolic and Elliptic Equations, Plenum Press, 1992 | Zbl

[18] Peter Poláčik; Eiji Yanagida On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., Volume 327 (2003) no. 4, pp. 745-771 | MR | Zbl

[19] Pavol Quittner; Philippe Souplet Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, 2019 | Zbl

[20] David H. Sattinger Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, 309, Springer, 1973 | MR | Zbl

[21] Walter A. Strauss Stable and unstable states of nonlinear wave equations, Nonlinear partial differential equations (Contemporary Mathematics), Volume 17, American Mathematical Society, 1983, pp. 429-441 (Proceedings of a Conference held at the University of New Hampshire, Durham/ New Hampshire, June 20-26, 1982, sponsored by the American Mathematical Society) | DOI | MR

[22] Salvador Villegas Asymptotic behavior of stable radial solutions of semilinear elliptic equations in $ℝ^{N}$ , J. Math. Pures Appl., Volume 88 (2007) no. 3, pp. 241-250 | DOI | MR | Zbl

[23] Salvador Villegas Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $ℝ^{n}$ , Commun. Pure Appl. Anal., Volume 10 (2011) no. 6, pp. 1817-1821 | DOI | MR | Zbl

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